General Modelling and Scaling Laws - IV - · PDF file• Direct representation of the full...
Transcript of General Modelling and Scaling Laws - IV - · PDF file• Direct representation of the full...
1
Data Analysis
Experimental Methods in Marine Hydrodynamics
Lecture in week 36
By Valentin Chabaud, post-doc in experimental methods, teaching assistant
for this course in previous years. On behalf of Pr. Sverre Steen.
Objectives of this lecture:
• Give you an overview of the most important methods of data analysis in
use in experimental marine hydrodynamics
• Give some examples of how to do data analysis using Matlab
Covers Chapter 10 in the Lecture Notes
2
Contents
• Typical types of tests:
– 1. Static tests
– 2. Decay tests
– 3. Regular wave tests
– 4. Irregular wave tests
• Pre-processing data
• Filtering
• Spectral Analysis
– Fourier transform
– Power Spectral Density (PSD)
• Example
3
Static tests• Tests expected to give a constant measured value
– Example: Ship resistance, propulsion and open water tests
• Only the mean value is used in further analysis
• Take care to avoid transient effects at start-up
• Notice that even for tests of stationary phenomena like ship resistance in
calm water, there will be oscillations in the signal
– To create a reliable average at least ten oscillations should be included in the
time window
– If the signal is polluted by oscillations at a single low frequency, an entire
number of oscillations should be included in the time window
60
70
80
90
100
110
120
130
140
10 15 20 25 30 35 40
Time [seconds]
Mo
del R
esis
tan
ce R
Tm [
N]
1.8
1.82
1.84
1.86
1.88
1.9
1.92
1.94
1.96
1.98
2
Carr
iag
e s
peed
[m
/s]
RTm
Speed
Transient part Stationary part
4
0 500 1000 1500 2000 2500 3000 3500 4000-3
-2
-1
0
1
2
3
0 2000 4000 6000 8000 10000 12000 14000 16000 18000-4
-3
-2
-1
0
1
2
3
4
X: 1.289e+04
Y: 2.389
Pre-processing data in Matlab (for all tests)
5
Effect of the record length
• Sinusoidal wave 𝑥 = 𝐴 𝑠𝑖𝑛 𝜔𝑡 + B• Theoretical mean value 𝜇𝑡ℎ = 𝐵
• Theoretical standard deviation 𝜎𝑡ℎ =𝐴
2
• The error expectation (the actual error depends on the initial phase) is 0
for entire numbers of cycles, else decreasing with number of cycles
6
• Select the start and end times tstart and tend
• Interpolate to make data uniformly sampled
• Clean data. Equipment limitations (especially in MC lab) lead
to:
– Erroneous data: Infinite (very large) or NaN (not a number).
– Missing data: 0. Can occur for a somewhat long period of time and
thus affects the results even if the mean value is small, even 0.
Pre-processing data in Matlab (for all tests)
t=tstart:dt:tend
x=interp1(t0,x0,t)
Raw data and time arrays
Selected time array
Uniformly sampledselected data
7
• The data can be cleaned by the function:
• Home made function. Tested on a limited number of time series only. Yet,
always check the results! Modifications and suggestions are welcome.
• Smoothen x using smooth(x,round(fs/fx)+1) if sampled at fx<fs (stair-
like signal)
• clean_data function is found in the Resource-section of the TMR7 webpage
and at the end of this presentation
xclean=clean_data(x’,CrtSTD,CrtCONV)
Cleaned data, row vector
Original data (uniformly sampled), row vector.
Iterative outlier criterion Convergence criterion
Play around with these criteria to get the desired result
Pre-processing data in Matlab (for all tests)
8
How clean_data works
𝑥𝑖 − 𝜇𝑥 ≥ 𝐶𝑟𝑡𝑆𝑇𝐷 ∗ 𝜎𝑥
ሶ𝑥𝑖 ≥ 𝐶𝑟𝑡𝑆𝑇𝐷 ∗ 𝜎 ሶ𝑥
ሶ𝑥𝑖 ≤𝜎 ሶ𝑥
10 ∗ 𝐶𝑟𝑡𝑆𝑇𝐷
Or
Or
Then2)
1) If
Replace 𝑥𝑖 by a linear interpolationof the nearest valid points
3) Recompute 𝜇𝑥 and 𝜎𝑥 and iterate until it has converged:
𝜎𝑥𝑛 − 𝜎𝑥𝑛−1𝜎𝑥𝑛−1
≤ 0.1
𝜇 : Mean value𝜎 : Standard deviation
Less error is induced by keeping corrupt points thansimply removing them!
Signal
Derivative
Tuning parameter- CrtSTD > 1- CrtSTD should be large when signal has
uneven amplitudes (if too small, cleaningcan affect valid parts of the signal)
Pre-processing data in Matlab (for all tests)
9
Decay Tests
• Model is oscillated and then
released, and response is
measured
• Provides information about
natural period and linear and
quadratic damping terms
• Very useful for lightly damped
degrees of freedom (system
dependent). For ships:
– Well suited: Natural period and
damping in roll, horizontal
motions
– Difficult, but possible: pitch
– Close to impossible: heave
Logaritmic decrement:
1
ln( )i
i
x
x
10
Analysis of
decay tests
1
ln( )i
i
x
x
02cr
p p
p M
0
0
22EQ
Cp M
The damping ratio:
For low damping ratios
(<0.2):
The logarithmic decrement:
Equivalent damping:
Λ = 2𝜋𝜉
11
Alternative analysis of decay tests
- fitting of equivalent theoretical system
http://www.ivt.ntnu.no/imt/courses/tmr7/resources/decay.pdf
12
• Use the function findpeaks to measure amplitudes and periods
Decay analysis in Matlab
[PKS,LOCS] = findpeaks(x,t)
T=diff(LOCS)
Gamma=log(PKS(1:end-1)/PKS(2:end))
Amplitudes Times
DecrementPeriods
13
Filtering
• Noise is undesirable in measurements
– Impairs accuracy at frequencies of interest by folding (aliasing) if
N𝑜𝑖𝑠𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 > 𝑁𝑦𝑞𝑢𝑖𝑠𝑡 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 =𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
2
– Impairs readability of time series
– Impairs accuracy of statistical properties of the signal (increases standard
deviation, modifies mean value for low frequency noise)
• Two types of noise
– Measurement noise: Unphysical noise specific to sensor (e.g. grid
frequency). Typically removed by low-pass filtering in the hardware, i.e.
prior to data acquisition.
– Process noise: Undesired dynamics of the system
• Transients: Decay of motion of undesired degrees of freedom. Typically low
frequency: High-pass filter (also removes mean value)
• Structural vibrations: excitation of off-interest eigenfrequencies of the system.
Typically high frequency: Low-pass filter
• Applied in the post-processing phase (by you!)
14
Filtering, cont.
How does it work?The gain of the filter’s transfer function attenuates some parts of the frequency content of the signal.
• In the frequency domain, no difference is made from 2 different processes having the same frequency
In order for filtering to be successful, undesired processes should have a distinct frequency content from that of the studied process.
• 𝐺 𝜔 must be continuous for the IFFT to exist.
The attenuation evolves gradually with the frequency. A sharp cut in the frequency content of a signal is not possible with low-order filters.
𝑥𝑓𝑖𝑙𝑡 𝑡 = 𝔉 −1 𝐺 𝜔 ∗ 𝔉 𝑥 𝑡 𝜔 (𝑡)
FFT of the signalGainFilteredsignal
IFFT
𝐺 𝜔 = 𝐻 𝑗𝜔
15
Various
FiltersAmplitude
Ideal characteristic
Real characteristic
Low pass filter
High pass filter
Band pass filter
Frequency "Notch" filter
16
Filtering, cont.
Digital Butterworth filters:
• Most commonly used filters for this kind of application. Hardware filter=Butterworth filter order 4
• Described by its transfer function (in practice in the discrete domain)
𝐻 𝑧 =𝑏 1 +𝑏 2 𝑧−1+⋯+𝑏(𝑛+1)𝑧−𝑛
1+𝑎 2 𝑧−1+⋯+𝑎(𝑛+1)𝑧−𝑛
• Designed in Matlab by
[b,a]=butter(order,wstar,’ftype’)
𝑤∗ =𝐶𝑢𝑡𝑜𝑓𝑓 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 (𝐻𝑧)
𝑁𝑦𝑞𝑢𝑖𝑠𝑡 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
Order of the filter
Normalized cutoff frequency(Or interval of frequencies for bandpass filter)
’low’ low-pass filter filters frequencies > cutoff freq.’high’ high-pass filter filters frequencies < cutoff freq.’bandpass’ band-pass filter filters frequencies outside the
cutoff freq interval.
1
2 ∗ 𝑡𝑖𝑚𝑒 𝑠𝑡𝑒𝑝
17
The filtering effect is best described by Bode diagrams of the
filter’s continuous transfer function
[b,a]=butter(order,wstar,’low’)
Figure()
Bode(d2c(tf(b,a,dt)))
Slope in gain reduction:
• = «filtering strength»
• Increasing with the order
• Increasing with frequency
(for a low-pass filter) from
cut-off frequency
The cut-off frequency
should be higher than the
undesired frequencies, but
lower than the frequencies of
interest.
Else the signal will be badly
filtered or the amplitude
attenuated!
Filtering induces
a phase shift in
the signal,
increasing with
order and
frequency
Cut-off frequency
Filtering, cont.
18
• A so-called “spectral gap” is needed for efficient filtering
= No energy in the spectrum around the cut-off frequency
If this is not the case, uncertainties will be introduced, take note of them!
• To avoid phase shift (improves readability in time domain plots), the time series are filtered first forward, then backward (symmetric or “zero-phase” filtering). This is not possible in real-time.
In Matlab:
xfilt=filtfilt(b,a,x)
Filtered data
Original data
(uniformly sampled)
Digital filter
coefficients
Filtering, cont.
19
Aims of analysis of regular wave tests
• Response amplitude
• Response amplitude operator (transfer function in
frequency domain)
– Gain = Response amplitude/wave amplitude
– Phase angle (between wave at reference location and response)
• Response frequencies
– In addition to wave frequency, nonlinear excitation of the natural
frequencies of the system
Reminder: Take care to leave out transient response at the
start of the time series!
20
Analysis procedure for regular wave tests
Time-domain visual
analysis
Frequency-domain visual
analysis
Computing Response
Amplitude
Computing Response
Phase
• Global assesment of the validity of the test
• First manual estimation of main period and amplitude
• Global assesment of the validity of the test
• Overview of response frequencies, estimation of the
noise level
• Design of possible filtering and checking filtering results
• Fourier series analysis. Gives both phase and amplitude
at a specified frequency (wave frequency) and its
harmonics.
• Compared with incident wave: gives the total RAO
(frequency-domain transfer function)
• Possible to extend to multiple frequencies (sum of sines)
• Divided by wave amplitude = Gain of RAO (Response
amplitude operator)
• From standard deviation of noise-free signal
• Average of peaks or 𝑚𝑖𝑛+𝑚𝑎𝑥
2are less accurate methods
Amp=sqrt(2)*std(x)
21
22
23
Fourier series Analysis
• Goal: Extracting the linear component of the response to regular waves
and deriving the gain and phase of the RAO at a given frequency (or
period).
• A periodic signal with period T can be fully described by an infinite sum
of harmonic components, called Fourier series:
0
0
0
0
1( )
2 2( )cos
2 2( )sin
T
T
k
T
k
a f t dtT
kta f t dt
T T
ktb f t dt
T T
With coefficients defined as:
0
1
2 2( ) cos sink k
k
kt ktf t a a b
T T
24
• Fit a Fourier Series to a time series in Matlab:1. Open the Curve Fitting Toolbox (>> cftool)
2. Choose Fourier from the model type list.
3. Use Fit Options and number of terms to control the fit.
4. Check if frequency (given by w) is correct
5. Retrieve the 𝑎1 and 𝑏1 coefficients (linear terms).
6. Calculate gain and phase:
• Do it for both incident wave and response. Compute Gain and
Phase of RAO by
Fourier series Analysis (cont.)
𝐺𝑎𝑖𝑛 = 𝑎12 + 𝑏1
2
𝑃ℎ𝑎𝑠𝑒 = 𝑎𝑡𝑎𝑛𝑏1𝑎1
𝐺𝑎𝑖𝑛𝑅𝐴𝑂 =𝐺𝑎𝑖𝑛𝑅𝑒𝑠𝑝𝑜𝑛𝑠𝑒
𝐺𝑎𝑖𝑛𝑊𝑎𝑣𝑒
𝑃ℎ𝑎𝑠𝑒𝑅𝐴𝑂 = 𝑃ℎ𝑎𝑠𝑒𝑅𝑒𝑠𝑝𝑜𝑛𝑠𝑒 − 𝑃ℎ𝑎𝑠𝑒𝑊𝑎𝑣𝑒
25
26
Example of accuracy of estimating
amplitude from st.dev. in regular waves
0
0.5
1
1.5
2
2.5
3
3.5
8002 8004 8006 8000 8002 8005 8001 8002 8003 8006
sqrt(2)*stdev(Waveheight 1) Fourier analysis Waveheight 1
sqrt(2)*stdev(Acc1_FP) Fourier analysis Acc1_FP1
RAO Acc1_FP sqrt(2)*stdev RAO Acc1_FP Fourier
27
Irregular wave tests
• Direct representation of the full scale sea condition
• Typically wanted results:
– Response spectra
– Response spectrum parameters:
• Spectral moments
• Standard deviation
• Peak period
– Response amplitude operator (RAO)
– Statistical results:
• Max and min values,
• Information about statistical distribution
• Extreme value statistics (extrapolation using the statistical distribution)
• Weibull plots etc.
28
Properties of stochastic processes
• Stationary: - Statistical properties constant with time
• Homogeneous: - Statistical properties constant in space
• Ergodic: - Time can be replaced by space as primary variable without
changing the statistical properties
• The wave environment is commonly assumed to be a stationary,
ergodic process
• This assumption greatly simplifies the analysis, and is a necessity for
all established analysis methods
• It is not exactly true in a towing tank
– Viscous damping: wave amplitude decreasing with distance to the wave
maker, transmission of energy between frequencies. Significant for long
towing tanks.
– Wave reflection: Non-homogeneous and non-stationary effects.
29
Autocorrelation function
0
1( ) ( ) ( )
T
xx
LimitR x t x t dt
T T
Figures from Newland (1984)
is the standard deviation and m is the average value
30
Cross-correlation function
0
1( ) ( ) ( )
T
xy
LimitR x t y t dt
T T
Figures from Newland (1984)
Cross-correlation function for
two sine waves with y(t) lagging
x(t) by an angle
31
Fourier transform
• In practice in the discrete (digital) domain: The continuous
and ergodic signal 𝑓(𝑡) is sampled (assumed uniformly) over
the record duration T at a rate 𝑓𝑠 (in Hz), giving the time
series 𝑓𝑘 with
𝑘 = 0,1,… ,𝑁 − 1; 𝑁 = 𝑇𝑓𝑠• The nth component of the Discrete Fourier Transform (DFT)
of 𝑓𝑘 reads
• 𝑓𝑘 can then be exactly retrieved by the Inverse DFT:
1i(2 n )
0
1ˆN
k N
n k
k
f f eN
𝑓𝑘 =
𝑛=0
𝑁−1
መ𝑓𝑛𝑒𝑖2𝜋𝑛𝑘𝑁
32
Fast Fourier Transform (FFT)
• FFT is a computer algorithm for calculation of DFT. It is a
core function of all digital data analysis.
• Conventional Discrete Fourier Transform (DFT) can also be
implemented in a computer
– DFT requires n2 multiplications
– FFT requires n·log(n) multiplications
– FFT is more accurate (due to fewer multiplications)
• FFT requires that N is a power of 2. Two solutions:
– Truncate the signal to the nearest, lower power of 2
– Augment the signal to the nearest, higher power of 2 by adding L zeros
(or samples equal to mean value if non zero-mean signal).
• Recommended for more accuracy.
• Corrections must be applied to the output
• Automatically done in the function fft in Matlab
33
Spectral Density
• Frequency-domain representation of the correlation.
– Fourier Transform of Autocorrelation = Power Spectral Density (PSD), key tool
to represent the frequency content (in terms of energy) of a signal
– Fourier Transform of Cross-correlation = Cross Spectral Density (CSD)
• Time series are in theory of infinite length and aperiodic. A direct FT is ill-
defined. The frequency content can only be obtained through FTs of
correlation functions instead, leading to PSDs and CSDs.
• In the discrete domain and over a finite record time T, the periodicity
requirement can be lifted. SDs are then in practice not computed using their
original definition (i.e. from autocorrelation), but from the product of FFTs
𝑆𝑓𝑔𝑛𝜔𝑛 = መ𝑓𝑛
∗ො𝑔𝑛
for each frequency 𝜔𝑛 =2𝜋𝑛
𝑇, 𝑛 = 0,1, … ,𝑁 − 1. 𝑓 = 𝑔 gives PSD
• From circular to linear frequency: 𝑆 𝑓 = 2𝜋𝑆 𝜔
Complex conjugate
34
Meaning of spectral moments
• The n’th moments of the spectrum is defined as:
• Standard deviation of response:
• Significant value of response:
• Average period of response:
• Average zero crossing period :
dSm n
n
0
)(
0m
1
01
m
mT
2
02
m
mT
031 4 mx
35
Accuracy and resolution of PSDs• The accuracy of the SD computation can be written as (Newland, 1984):
𝜎
𝜇≅
1
2𝑛 + 1
Standard deviation and
mean value of PSD at
frequency f in a set of
records of length T
Moving average smoothing with the n
previous and n following frequencies
• n determines the smoothness of the
spectrum. Increasing smoothness increases
accuracy, but decreases resolution.
• Assuming1
𝑇sufficiently small wrt f, the
accuracy is not dependent on T, nor on f !
• Increasing T increases resolution and
enables the study of lower frequencies.
• The effect of the record length on the
standard deviation of sine waves can be
applied to frequency components of PSDs
Assuming 1/T is much lower (>5 -10 times) than f
36
• Many functions, hard to tune because complex underlying mathematics
• psd_fft is a home made function computing the PSD directly from
the Fourier transform.
• Designed from: An introduction to Random Vibrations and Spectral
Analysis, by D.E. Newland and The Mathworks website
http://se.mathworks.com/help/signal/ug/psd-estimate-using-fft.html
• psd_fft.m is found in the Resource-section of the TMR7 webpage
and in this presentation (next slide)
PSD in Matlab
Sxx=psd_fft(x,Ns,f,fs)
Sampling frequency (Hz)
Frequencies at which you want the
PSD to be computedPSD (in 𝑢𝑛𝑖𝑡(𝑥)2. 𝐻𝑧−1)
Signal (cleaned,
uniformly sampled)
Number of points in moving average = 2𝑛 + 1
37
function [S,Sraw]=psd_fft(x,Ns,f,fs)
%Calculate PSD from raw fft and smoothing. From Newland: “An introduction to
% random vibrations and spectral analysis” and The Nathworks website
% http://se.mathworks.com/help/signal/ug/psd-estimate-using-fft.html
%x: signal
%Ns: Number of points in moving average (=2*n+1, odd number)
%f: desired output frequencies
%fs: sampling frequency
%S: PSD @ frequencies f
%Sraw: Structure with field S=PSD and field f=frequencies as defined by fft
Nt=floor(size(x,1)/2)*2;
x=x(1:Nt,:);
dt=1/fs; %Step size
T=Nt*dt; %Record length
S=fft(x); %Compute dft by fft. fft specificities (added zeroes) are handled
% internally.
S=2*dt/Nt*abs(S(1:Nt/2+1,:)).^2; %Compute PSD: one-sided (multiply by 2),
% distributed (divide by sampling freq) average of fft (divide by number of
% points) squared
fS = 1/dt*(0:(Nt/2))/Nt; %Output frequencies, up to number of points/2 (higher
% frequencies only show a folded version of low frequencies)
for i=1:size(x,2)
S(:,i)=[S(1,i)/2;smooth(S(2:end,i),Ns)]; %Smoothing by moving average
end
Sraw.S=S;
Sraw.f=fS;
S=interp1(fS,S,f); %Interpolation to desired output frequencies
psd_fft.m
38
pwelch is the standard built-in Matlab function for PSD calculation.
• Default values often lead to inaccurate results.
• Excessively computationally demanding for long time series
• More accurate than psd_fft for short time series, because moving average smoothing diffuses uncertainties of low frequencies onto frequencies of interest.
Sxx=2*pwelch(x,Window,Noverlap,f,fs)
Number of overlapping samples between
windows. Does not have a big influence.
Window/10 is a good start.
The signal is segmented into «windows». The FFT is
computed segment by segment which are then assembled
to give the PSD.
The broader the window, the finer the spectrum. The
narrower, the smoother. Adjust it to get a readable yet
accurate spectrum (Use values from NFFT/2 to NFFT/10).
Change from two-sided to
one-sided PSD
PSD in Matlab, cont.
39
Transfer function in irregular waves
(equivalent to RAO in regula waves)
)(
)()(
2
xx
yy
S
SH
𝐻 𝜔 =𝑆𝑥𝑦 𝜔
𝑆𝑥𝑥 𝜔
Phase can be obtained
from CSD:
Magnitude:
40
Summary
• Static tests and pre-processing
– The valid window of the time series to be analyzed must be sufficiently long
– The data must be cleaned and uniformly sampled
• Filtering
– Filter design in the frequency domain
– Need for a spectral gap
– Zero-phase filtering
• Regular wave tests
– The amplitude should be calculated through the standard deviation of filtered signals
– If the phase is desired, use Fourier Series
• Irregular wave tests
– Spectral densities based on the Fourier transform are used for frequency domain analysis
– Appropriate smoothing should be applied
41
Example of post-processing with Matlab:
Irregular wave elevation
Generated from JONSWAP spectrum.
The following is artificially added:
– Erroneous and missing data
– Measurement noise
– Transients
– Mean offset
42
Example cont. : Matlab script
load(‘data.mat',‘x‘,’time’) %Load wave elevation and time from file
duration=200;
dt=0.1;
t=0:dt:duration;
Nt=length(t);
xint=interp1(time,x,t); %Interpolate data
xclean=clean_data(xint,3,0.001); %Clean data
cutoff=[0.3 4]/(2*pi); %Cut-off frequencies
fnyq=1/(2*dt); %Nyquist frequency
[b,a]=butter(4,cutoff/fnyq,'bandpass'); %Get filter coefficients
xfilt=filtfilt(b,a,xclean); %Zero-phase filtering
df1=0.01;
df2=0.1;
f1=0.01:df1:0.99; %Small frequency step for low frequencies
f2=1:df2:10; %Large frequency step for high
frequencies
f=[f1 f2];
Sxx=psd_fft(xint-mean(xint),10,f,1/dt); %PSD of unfiltered data
Sxx_filt=psd_fft(xfilt-mean(xfilt),10,f,1/dt); %PSD of filtered data
figure(1)
plot(t,[x0 xint xclean xfilt]) %x0: original data generated
from JONSWAP
figure(2)
plot(w,jonswap,f*2*pi,Sxx/(2*pi),f*2*pi,Sxx_filt /(2*pi))
43
Example cont. : time and frequency domain plots
Large spectral gaps
allowing efficient filtering
of the noise
Noise
Uncertainties due to
short time series
Transients
Uncomplete spectral gap:
slightly uncertain filtering
of the transients
Cut-off frequenciesOutlier
Period of missing data
Band-pass filtering removes
hig and low (including offset
= 0 rad/s) frequencies
44
Questions?
Teaching assistant:
Office D2.235
About this course:
Office G2.130
45
while abs((std(x)-std_prev)/std_prev)>CrtCONV
flag=0;
ind=[];
for i=1:length(x)
if abs(x(i)-mx)>sx*CrtSTD ||
abs(d(i))>sd*CrtSTD || abs(d(i))<sd/CrtSTD*0.1
if flag==0
flag=1;
ind=[ind;[i 0]];
end
else
if flag==1
ind(end,2)=i;
flag=0;
end
end
end
if(ind(end,end))==0
ind(end,end)=length(x);
end
y=[ones(N,1)*x(1);x;ones(N,1)*x(end)];
for i=1:size(ind,1)
inttot=(1:length(y))';
intrem=ind(i,1)+N:ind(i,2)+N;
intfit=setdiff(inttot,intrem);
z=y(intfit);
% f = fit(intfit, z,
'smoothingspline','SmoothingParam', 0.1);
% y(intrem)=feval(f,intrem);
y(intrem)=interp1(intfit,y(intfit),intrem);
x=y(N+(1:length(x)));
end
std_prev=std(x);
end
x=x';
clean_data.mfunction x=clean_data(data,CrtSTD,CrtCONV)
%Written by Valentin Chabaud. v3 - August
2015
%Removes erroneous values and outsiders
from time series
x=data';
sx=std(x);
mx=mean(x);
d=diff(x);
sd=std(d);
d=[d;d(end)];
% figure(3)
% plot([data';d])
std_prev=std(x)/CrtSTD;
N=10;
46
Statistical distributions
• The probability distribution function, P(x), is the
probability that a general value of the process x(t) is less
than or equal to the value of x
• The probability density function:
• The probability that a<x(t)<b is given by the probability
density function such that:
( ) ( ( ) )P x P x t x
( )( )
dP xp x
dx
( ( ) ) ( )
b
a
P a x t b p x dx
47
Probability distributions used in the
study of wave generated responses
• The distribution of the process itself, e.g. the distribution
of the wave elevation x(t) and the measured response y(t)
Gaussian distribution
• The distribution of amplitudes; e.g. distribution of the
wave amplitudes xA and measured response amplitudes, yA
in the tests.
Rayleigh distribution
48
Rayleigh distribution of amplitudes
• Follows from the assumption that the elevation itself is
Gaussian
• The cumulative distribution:
• Here is the mean or expected value of x(t) defined as:
• is the variance of x(t), defined as:
2
1( ) 1 exp
2
X
X
xP x
( )X E x xp x dx
2 22 2
X X XE x E x
49
Rayleigh distribution
• For a measured time series with
N samples the mean value and
the variance are calculated as:
1
1 N
X i
i
xN
2
1
1( )
1
N
X i X
i
xN
50
Non-linear response
• The response y(t) follows a Rayleigh distribution only if it
is a linear function of the wave elevation x(t)
• To describe non-linear response it is common to use the
more general Weibull distribution:
– G=2 gives the Rayleigh distribution
– G=1 gives the Exponential distribution
1( ) 1 exp
G
A XA
xP x
G
51
)P(x1lnln A
Weibull plots
Horisontal axis: response amplitude yA Horisontal axis: response amplitude yA
P(x
A)-
axis
plo
tted a
s
52
Significant values
• Significant maxima:
– the mean of the highest one-third of the crest-to-zero values of xA,
• Significant minima:
– the mean of the highest one-third of the trough-to-zero values of xA,
• Significant double amplitude:
– mean of the highest one-third of the maximum to minimum values of xA
53
Maximum/Minimum Values
• Maximum Value:
– Measured maximum value in the record
• Minimum Value:
– Measured minimum value in the record
• Largest double amplitude:
– Largest measured crest to trough value in the record
56
Examples of special analyses:
• Slamming
• Sea sickness incidence
• Ventilation and air injection to waterjets
57
Slamming analysis
• Definition of slamming threshold value(s)
– Typically 50 kPa, but depends heavily on context
• Counting (automatically) the number of slams above
different threshold levels
• Detailed analysis of the time series of each slam reveals
properties of the slam, the transducer and the model
dynamic response
58
61
Sea sickness incidence
• Estimation of sea sickness incidence is based on:
– Measurement of motions and accelerations of the model/ship
– Measurement of motion sickness incidence MSI (percentage of
people vomiting) to vertical accelerations of different frequency,
amplitude and duration
• Empirical relations of motion sickness incidence (MSI) as
function of frequency, RMS amplitude, and duration
available in ISO standard ISO 2631 1-4
NBDLCABIN
HEAVE
GUIDE
RAIL
GUIDE
RAIL
ROLL
AXIS
PITCH
AXIS
+TILT
TABLE
MOVING
A-FRAME
62